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A007209
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Number of partitions of n into parts of sizes {a( )} is a(n).
(Formerly M0315)
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1
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1, 1, 2, 2, 4, 5, 7, 9, 12, 16, 20, 25, 32, 39, 49, 58, 73, 86, 105, 123, 149, 175, 207, 241, 284, 331, 385, 444, 515, 592, 682, 777, 894, 1015, 1160, 1310, 1492, 1683, 1903, 2140, 2412, 2708, 3037, 3395, 3801, 4239, 4730, 5254, 5852, 6489, 7204, 7965, 8823, 9741
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OFFSET
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0,3
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
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FORMULA
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G.f. A(x) = 1 / ((1 - x) * Product_{k>2} (1 - x^a(k))).
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EXAMPLE
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G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 12*x^8 + 16*x^9 + ...
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MAPLE
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a:= proc(n) option remember; local x, j, i; `if`(n<6, [1, 1, 2, 2, 4, 5][n+1], coeff(series(1 /mul(1-x^i, i=[{seq(a(j), j=1..n-1)}[]]), x, n+1), x, n)) end: seq(a(n), n=0..60); # Alois P. Heinz, Apr 27 2009
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MATHEMATICA
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a[0] = a[1] = 1; a[2] = a[3] = 2; a[4] = 4; a[5] = 5; a[n_] := a[n] = SeriesCoefficient[ Series[ 1/((1 - x)*Product[ 1 - x^a[k], {k, 3, n-1}]), {x, 0, n}], n]; Table[a[n], {n, 0, 53}] (* Jean-François Alcover, Dec 09 2011 *)
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PROG
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(PARI) {a(n) = local(A, k); A = Ser([ 1, 1, 2, 2, 4, 5]); while( n > (k = #A - 1), A = 1 / (1 - x) / prod( i=3, k, 1 - x^polcoeff(A, i), 1 + x^2 * O(x^ polcoeff( A, k))) ); polcoeff( A, n)}; /* Michael Somos, Aug 08 2011 */
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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